Annals of Mathematics


Quasi-isometry
invariance of group
splittings

By Panos Papasoglu

Annals of Mathematics, 161 (2005), 759–830
Quasi-isometry invariance
of group splittings
By Panos Papasoglu
Abstract
We show that a finitely presented one-ended group which is not commen-
surable to a surface group splits over a two-ended group if and only if its Cayley
graph is separated by a quasi-line. This shows in particular that splittings over
two-ended groups are preserved by quasi-isometries.
0. Introduction
Stallings in [St1], [St2] shows that a finitely generated group splits over a
finite group if and only if its Cayley graph has more than one end. This result
shows that the property of having a decomposition over a finite group for a
finitely generated group G admits a geometric characterization. In particular
it is a property invariant by quasi-isometries.
In this paper we show that one can characterize geometrically the prop-
erty of admitting a splitting over a virtually infinite cyclic group for finitely
presented groups. So this property is also invariant by quasi-isometries.
The structure of group splittings over infinite cyclic groups was understood
only recently by Rips and Sela ([R-S]). They developed a ‘JSJ-decomposition
theory’ analog to the JSJ-theory for three manifolds that applies to all finitely
presented groups. This structure theory underlies and inspires many of the

geometric arguments in this paper. A different approach to the JSJ-theory for
finitely presented groups has been given by Dunwoody and Sageev in [D-Sa].
Their approach has the advantage of applying also to splittings over Z
n
or
even, more generally, over ‘slender groups’.
Bowditch in a series of papers [Bo 1], [Bo 2], [Bo 3] showed that a one-
ended hyperbolic group that is not a ‘triangle group’ splits over a two-ended
group if and only if its Gromov boundary has local cut points. This charac-
terization implies that the property of admitting such a splitting is invariant
under quasi-isometries for hyperbolic groups. Swarup ([Sw]) and Levitt ([L])
contributed to the completion of Bowditch’s program which led also to the
solution of the cut point conjecture for hyperbolic groups.
760 PANOS PAPASOGLU
To state the main theorem of this paper we need some definitions: If Y is
a path-connected subset of a geodesic metric space (X, d) then one can define
a metric on Y , d
Y
, by defining the distance of two points to be the infimum of
the lengths of the paths joining them that lie in Y .Aquasi-line L ⊂ X is a
path-connected set such that (L, d
L
) is quasi-isometric to R and such that for
any two sequences (x
n
), (y
n
) ∈ L if d
L
(x

n
,y
n
) →∞then d(x
n
,y
n
) →∞.
We say that a quasi-line L separates X if X − L has at least two compo-
nents that are not contained in any finite neighborhood of L.
With this notation we show the following:
Theorem 1. Let G be a one-ended, finitely presented group that is not
commensurable to a surface group. Then G splits over a two-ended group if
and only if the Cayley graph of G is separated by a quasi-line.
This easily implies that admitting a splitting over a two-ended group is a
property invariant by quasi-isometries. More precisely we have the following:
Corollary. Let G
1
be a one-ended, finitely presented group that is not
commensurable to a surface group. If G
1
splits over a two-ended group and G
2
is quasi-isometric to G
1
then G
2
splits also over a two-ended group.
We note that a different generalization of Stalling’s theorem was obtained
by Dunwoody and Swenson in [D-Sw]. They show that if G is a one-ended

group, which is not virtually a surface group, then it splits over a two-ended
group if and only if it contains an infinite cyclic subgroup of ‘codimension 1’.
We recall that a subgroup J of G is of codimension 1 if the quotient of the
Cayley graph of G by the action of J has more than one end. The disadvantage
of this characterization is that it is not ‘geometric’; in particular our corollary
does not follow from it. On the other hand [D-Sw] contains a more general
result that applies to splittings over Z
n
. Our results build on [D-Sw] (in fact
we only need Proposition 3.1 of this paper dealing with the ‘noncrossing’ case).
The idea of the proof of Theorem 1 can be grasped more easily if we
consider the special case of G = Z
3

Z
Z
3
. One can visualize the Cayley graph
of G as a tree in which the vertices are blown to copies of Z
3
and two adjacent
vertices (i.e. Z
3
’s ) are identified along a copy of Z. Now the copies of Z
3
are
‘fat’ in the sense that they cannot be separated by a ‘quasi-line’. The Cayley
graph of G on the other hand is not fat as it is separated by the cyclic groups
corresponding to the edge of the splitting. This is a pattern that stays invariant
under quasi-isometry: A geodesic metric space quasi-isometric to the Cayley

graph of G is also like a tree; the vertices of the tree are ‘fat’ chunks of space
that cannot be separated by ‘quasi-lines’ and two adjacent such ‘fat’ pieces are
glued along a ‘quasi-line’.
The proof of the general case is along the same lines but one has to take
account of the ‘hanging-orbifold’ vertices of the JSJ decomposition of G.
QUASI-ISOMETRY INVARIANCE OF GROUP SPLITTINGS
761
The main technical problem is to show that when the Cayley graph of a
group is separated by a quasi-line then ‘fat’ pieces do indeed exist. To be more
precise one has to show that if any two points that are sufficiently far apart are
separated by a quasi-line then the group is commensurable to a surface group.
For this it suffices to show that the Cayley graph of G is quasi-isometric to
a plane. So what we are after is an up to quasi-isometry characterization of
planes.
The first such characterization was given by Mess in his work on the Seifert
conjecture ([Me]). There have been some more such characterizations obtained
recently by Bowditch ([Bo 4]), Kleiner ([Kl]) and Maillot ([Ma]).
The characterization that we need for this work is quite different from the
previous ones. ‘Large scale’ geometric problems are often similar to topological
problems. Our problem is similar to the following topological characterization
of the plane:
Let X be a one-ended, simply connected geodesic metric space such that
any two points on X are separated by a line. Then X is homeomorphic to a
plane.
We outline a proof of this in the appendix. It is based on the classic
characterization of the sphere given by Bing ([Bi]).
The proof of the large scale analog to this runs along the same line but is
more fuzzy as a quasi-prefix has to be added to the definitions and arguments.
Although we could carry out the analogy throughout the proof, we simplify
the argument in the end using the homogeneity of the Cayley graph. We use

in particular Varopoulos’ inequality to conclude in the nonhyperbolic case and
the Tukia, Gabai, Casson-Jungreis theorem on convergence groups ([T], [Ga],
[C-J]) to deal with the hyperbolic case.
The topological characterization of the plane presented in the appendix is
quite crucial for understanding the quasi-isometric characterization of planar
groups used here. We advise the reader to understand the topological argument
of the appendix before reading its ‘large scale’ generalization (Sections 1–3
of this paper). A principle underlying this work is that many topological
results have, when reformulated appropriately, large scale analogs. Both the
proofs and the statements of these analogs can be involved but this is more
due to the difficulty of ‘translation’ to large scale than genuine mathematical
difficulty. We hope that the statement and proof of Proposition 2.1 offers a
good introduction to ‘translating’ from topology to large scale.
We explain now how this paper is organized: In Section 2 we show
(Prop. 2.1) that if a quasi-line L separates a Cayley graph in three pieces
then points on L cannot be separated by quasi-lines. We state below Propo-
sition 2.1 (we state it in fact in a slightly different, but equivalent, way in
Section 2):
762 PANOS PAPASOGLU
Proposition 2.1. Let X be a locally finite simply connected complex and
let L be a quasi-line separating X, such that X − L has at least three distinct
essential connected components X
1
,X
2
,X
3
.IfL
1
is another quasi-line in X

then L is contained in a finite neighborhood of a single component of X − L
1
.
We call a component X
i
essential if X
i
∪ L is one-ended. We remark
that the proposition above is similar to the following topological fact: Let X
be the space obtained by gluing three half-planes along their boundary line.
Then points on the common boundary line of the three half-planes cannot be
separated by any line in X. We will actually need a stronger and somewhat
less obvious form of this that is proved in Lemma A.1 of the appendix. The
proof of Proposition 2.1 is a ‘large scale’ version of the proof of Lemma A.1.
Proposition 2.1 is used in Section 3 to give a new ‘quasi-isometric’ char-
acterization of planar groups:
Theorem. Let G be a one-ended finitely presented group and let X = X
G
be a Cayley complex of G. Suppose that there is a quasi-line L such that for any
K>0 there is an M>0 such that any two points x, y of X with d(x, y) >M
are K-separated by some translate of L, gL (g ∈ G). Then G is commensurable
to a fundamental group of a surface.
The theorem above is in fact slightly weaker than Theorem 3.1 that we
prove in Section 3. The proof of this is a ‘large scale’ version of the proof of
the main theorem of the appendix:
Theorem A. Let X be a locally compact, geodesic metric space and let
f : R
+
→ R
+

be an increasing function such that lim
x→0
f(x)=0.IfX
satisfies the following three conditions then it is homeomorphic to the plane.
1) X is one-ended.
2) X is simply connected.
3) For any two points a, b ∈ X there is an f-line separating them.
We refer to the appendix for the definition of f -lines which is somewhat
technical. To make sense of the theorem above think of f-lines as proper lines,
i.e. homeomorphic images of R in X.
It turns out that to carry out our proof we need a stronger version of
Theorem 3.1 proved in Section 4. It says roughly that if G is not virtually
planar then its Cayley graph has an unbounded connected subset S such that
no two points on S can be separated by a quasi-line (Theorem 4.1). We call
such subsets solid. In the example G = Z
3

Z
Z
3
this subset corresponds to a
Z
3
-subgroup.
The proof of Theorem 4.1 is based on the homogeneity of the Cayley
graph of G. The characterization theorem of virtual surface groups given in
Section 4 allows us to pass from large scale geometry to splittings. The idea is
QUASI-ISOMETRY INVARIANCE OF GROUP SPLITTINGS
763
that maximal unbounded solid sets are at finite Hausdorff distance from vertex

groups of the JSJ-decomposition of G. This is easier to show when these sets
are ‘big’, i.e. they are not themselves quasi-lines. This is the case for example
if G = Z
3

Z
Z
3
. If on the other hand G is, say, a Baumslag-Solitar group then
all solid sets in its Cayley graph are quasi-lines.
In Section 5 we show (Proposition 5.3) that solid subsets correspond to
subgroups when they are not quasi-isometric to quasi-lines. In fact they are
vertex groups for the Bass-Serre tree corresponding to a splitting of G over a
two-ended group. We prove then Theorem 1, in case there are solid subsets of
X which are not quasi-lines, by applying [D-Sw].
In Section 6 we deal with the ‘exceptional’ case in which all solid subsets
are quasi-lines. This is split in several cases. We show depending on the case
either directly that G splits over a two ended subgroup by applying again
[D-Sw], or that G admits a free action on an R-tree, in which case we conclude
by Rips’ theory ([B-F]). This completes the proof of Theorem 1.
We note that Section 6 is essentially self-contained. It does not require
the technical results of the appendix and their large scale analogs. It could be
read directly after the preliminaries and the definition of solid sets in Section 4
as it offers a good illustration of how one can derive splitting results from
a mild geometric assumption which is valid in many cases (for example this
assumption holds for Baumslag-Solitar groups).
In Section 7 we show that JSJ decompositions are invariant under quasi-
isometries. More precisely we have the following:
Theorem 7.1. Let G
1

,G
2
be one-ended finitely presented groups, let
Γ
1
, Γ
2
be their respective JSJ-decompositions and let X
1
,X
2
be the Cayley
graphs of G
1
,G
2
.
Suppose that there is a quasi-isometry f : G
1
→ G
2
. Then there is
a constant C>0 such that if A is a subgroup of G
1
conjugate to a ver-
tex group, an orbifold hanging vertex group or an edge group of the graph of
groups Γ
1
, then f(A) contains in its C-neighborhood (and it is contained in the
C-neighborhood of ) respectively a subgroup of G

2
conjugate to a vertex group,
an orbifold hanging vertex group or an edge group of the graph of groups Γ
2
.
It is an interesting question whether Theorem 1 is true for finitely gen-
erated groups in general. The existence of a characterization like the one in
Theorem 1 was posed as a question by Gromov in the 1996 Group Theory
Conference in Canberra.
I would like to thank A. Ancona, F. Leroux, B. Kleiner, P. Pansu and
Z. Sela for conversations related to this work. I am grateful to David Epstein
for many stimulating discussions on plane topology and for his comments on
an earlier version of this paper.
764 PANOS PAPASOGLU
1. Preliminaries
A metric space X is called a geodesic metric space if for any pair of points
x, y in X there is a path p joining x, y such that length(p)=d(x, y). We
call such a path a geodesic. A geodesic triangle in a geodesic metric space X
consists of three geodesics a, b, c whose endpoints match. A geodesic metric
space X is called (δ)-hyperbolic if there is a δ ≥ 0 such that for all triangles
a, b, c in X any point on one side is in the δ-neighborhood of the two other
sides. If G is a finitely generated group then its Cayley graph can be made
a geodesic metric space by giving to each edge length 1. A finitely generated
group is called (Gromov) hyperbolic if its Cayley graph is a (δ)-hyperbolic
geodesic metric space. A path α :[0,l] → X is called a (K, L)-quasigeodesic
if there are K ≥ 1,L ≥ 0 such that length(α|
[t,s]
) ≤ Kd(α(t),α(s)) + L for all
t, s in [0,l]. In what follows we will always assume paths to be parametrized
with respect to arc length. A (not necessarily continuous) map f : X → Y is

called a (K, L) quasi-isometry if every point of Y is in the L-neighborhood of
the image of f and for all x, y ∈ X
1
K
d(x, y) − L ≤ d(f(x),f(y)) ≤ Kd(x, y)+L.
Definition 1.1. Let X, Y be metric spaces. A map f : X → Y is called
uniformly proper if for every M>0 there is an N>0 such that for all A ⊂ Y ,
diam(A) <M⇒ diam(f
−1
(A)) <N.
We remark that this notion is due to Gromov. In [G2] embeddings that
are uniformly proper maps are called uniform embeddings. It is easy to see
that the inclusion map of a finitely generated group H in a finitely generated
group G is a uniformly proper map (where G and H are given the word metric
corresponding to some choice of system of generators for each).
In what follows we consider R as a metric space.
Definition 1.2. Let X be a metric space. Let L : R → X be a one-to-
one, continuous map. We suppose that L is parametrized with respect to arc
length (i.e. length(L[x, y]) = d(x, y) for all x, y). We then call L a line if it is
uniformly proper.
There is a distortion function associated to L, D
L
: R
+
→ R
+
defined as
follows:
D
L

(t) = sup{diam(L
−1
(A)), where diam (A) ≤ t}.
We often identify L with its image L(R) and write L ⊂ X.Ifa = L(a

),b =
L(b

) are points in L, we denote by [a, b] the interval between a, b in L (so
[a, b]=L([a

,b

])), and by |b − a| the length of this interval. We write a<bif
a

<b

.Ift ∈ R we denote by a − t the point L(a

− t).
QUASI-ISOMETRY INVARIANCE OF GROUP SPLITTINGS
765
Definition 1.3. Let X be a metric space. We call L ⊂ X a quasi-line if
L is path connected and if there is a line L

⊂ L and N>0 such that every
point in L can be joined to L

by a path lying in L of length at most N.

One can also define quasi-lines as follows: Let L ⊂ X be a path connected
subset of X. We consider L as a metric space by defining the distance of two
points in L to be the length of the shortest path in L joining them (or the
infimum of the lengths if there is no shortest path). Then L is a quasi-line if:
i) L is quasi-isometric to R.
ii) L is uniformly properly embedded in X.
We say that L ⊂ X is an (f,N)-quasi-line, where f is a proper increasing
function, f : R
+
→ R
+
,ifL lies in the N-neighborhood of a line L

and
D
L

(t) ≤ f(t) for all t>0.
Suppose that the quasi-line L lies in the N-neighborhood of a line L

.We
define then a map a ∈ L → a

∈ L

where d(a, a

) ≤ N. Clearly there are many
possible choices for this map; we choose one such map arbitrarily. If a, b ∈ L
we define the interval between a, b in L as follows:

[a, b]
L
= {x ∈ L : d(x, [a

,b

]) ≤ N}.
Clearly this depends on the map a → a

. It is convenient to talk about the
‘length’ of the intervals of L. We define length([a, b]
L
) = length([a

,b

]).
We similarly define a partial order on L by a<bif and only if a

<b

.If
t ∈ R and a ∈ L then a + t is by definition the point a

+ t ∈ L

⊂ L. In what
follows when we write that a quasi-line L is in the N-neighborhood of a line
L


we will tacitly imply that a map a → a

is also given.
We will use throughout the notation for lines corresponding to quasi-lines,
so if L is an (f,N)- quasi-line we will denote by L

the line corresponding to
L (see Def. 1.3).
The following definition is abusive but useful:
Definition 1.4. Let X be a metric space and let L be a quasi-line in X.
We call a connected component of X − L, Y , essential if Y ∪ L is one-ended.
We say that a quasi-line L separates X,ifX − L has at least two essential
connected components and there is an M>0 such that every nonessential
component of X − L is contained in the M-neighborhood of L.
The following proposition shows that our definition is equivalent to a
weaker and more natural notion of separation.
Proposition 1.4.1. Let X be a Cayley graph of a finitely presented
one-ended group G and let L be an (f, N)-quasi-line such that for every n>0
there are x, y ∈ X such that d(x, L) >n,d(y, L) >nand x, y lie in different
components of X − L. Then there is an (f,N)-quasi-line L
1
that separates X.
766 PANOS PAPASOGLU
Proof. We show first that there is an (f,N)-quasi-line L
0
such that X −L
0
has at least two essential components. For any r>0 sufficiently big and for
any t ∈ L there is a path in X − L joining the two infinite components of
L − B

t
(r). Without loss of generality we can assume that this path (except
its endpoints) is contained in a single component of X − L. We call this path
p(t, r).
Since X is locally finite and G is finitely presented we can assume that
there are a t ∈ L and an r
0
> 0 such that p(t, r) lies for every r>r
0
in the
same component of X − L,sayC. Since G is one-ended C is clearly essential.
By our hypothesis we have that there is a sequence y
n
such that d(y
n
,L) >n
and y
n
/∈ C. Let q
n
be a geodesic joining y
n
to L with endpoint t
n
∈ L and
such that length(q
n
)=d(y
n
,L). Let us denote by T

n
the union L ∪ p
n
.We
then pick g
n
∈ G such that g
n
t
n
= t and next consider the sequence g
n
T
n
.Itis
clear that there is a subsequence of g
n
, denoted for convenience also by g
n
,so
that g
n
T
n
converges on compact sets to a union L
0
∪ p where L
0
is a quasi-line
and p is an infinite half geodesic lying in the same component of X − L

0
.By
passing if necessary to a subsequence we can ensure that X − L
0
has at least
one essential component disjoint from p.
Indeed, note that there is a sequence r
n
∈ N, r
n
→∞, such that for any
x ∈ L there are simple paths p(x, n) with the following properties (see Fig. 1):
1. p(x, n) is contained in
¯
C and p(x, n) joins the two unbounded components
of L − B
x
(r
n
).
2. p(x, n) ∩ B
x
(r
n
)=∅ and p(x, n) ⊂ B
x
(r
n+1
).
3. There is a path q(x, n) contained in B

x
(r
n+2
) ∩ C joining p(x, n)to
p(x, n + 1).
By passing to a subsequence we can ensure that for every k>0 the following
holds: For every n>k,
g
n
(p(t
n
,n) ∪ q(t
n
,n)) = g
k
(p(t
k
,k) ∪ q(t
k
,k)).
This clearly implies that X − L
0
has at least one essential component disjoint
from p.
Let C
1
be the component of X − L
0
containing p. Suppose that C
1

∪ L
0
is not one-ended. Then there is a compact K such that (C
1
∪ L
0
) − K is
two-ended and there is an infinite component of L
0
− K,sayL
+
0
, such that
C
1
∪ L
+
0
is one-ended. We can then pick x
n
∈ L
+
0
, x
n
→∞and h
n
∈ G such
that h
n

x
n
= t. By passing, if necessary, to a subsequence we can assume that
h
n
L
0
converges on compact sets to a quasi-line, denoted, to simplify notation,
still by L
0
. As before we can ensure that X − L
0
has at least two essential
components.
QUASI-ISOMETRY INVARIANCE OF GROUP SPLITTINGS
767
.
p(t, n +1)
p(t, n)
t
t
n
t
n+1
y
n
y
n+1
Figure 1
We have shown therefore that there is a quasi-line L

0
such that X −L
0
has
at least two essential components. Note also that if L is an (f,N)-quasi-line
L
0
is also an (f,N)-quasi-line.
Showing that there is a quasi-line satisfying the conclusion of the propo-
sition is proved in the same way: Suppose that there is a sequence z
n
∈ X
such that d(z
n
,L
0
) >nfor all n ∈ N and such that the z
n
do not belong to
any essential component of X − L
0
. We then pick geodesics q
n
joining z
n
to
L with length(q
n
)=d(z
n

,L
0
) and we pick k
n
∈ G such that k
n
z
n
= e (where
e is a fixed vertex). We show as above that there is a subsequence of k
n
L
0
converging on compact sets to a quasi-line L
1
such that X − L
1
has at least
three essential components.
We continue in the same way to produce new quasi-lines. It is clear that
this procedure terminates and produces a quasi-line, which we call, as in the
conclusion of the lemma, L
1
, such that if z
n
∈ X satisfies that d(z
n
,L
0
) →∞

then almost all z
n
lie in essential components of X − L
1
.
We remark that the procedure terminates because given f,N there is an
M>0 such that for any (f,N)-quasi-line L, X − L has less than M essential
components.
Remark 1.4.2. We can show in the same way the following slightly
stronger result: Let X be a Cayley graph of a finitely presented one-ended
group G and let L
n
be a sequence of (f,N)-quasi-lines such that for every
n>0 there are x, y ∈ X such that d(x, L
n
) >n,d(y, L
n
) >nand x, y lie in
different components of X −L
n
. Then there is a quasi-line L that separates X.
It is clear that a finite neighborhood of a quasi-line is itself a quasi-line.
The next proposition strengthens Proposition 1.4.1 to neighborhoods of quasi-
lines.
768 PANOS PAPASOGLU
Proposition 1.4.3. Let X be a Cayley graph of a finitely presented
one-ended group G.Ifan(f,N)-quasi-line L separates X then there is an
(f,N)-quasi-line L
0
such that for every r>0,N

r
(L
0
) separates X.
Proof. We define a sequence of (f,N)-quasi-lines L
n
(n>0) such that
N
k
(L
n
) separates X for all k ≤ n.IfN
1
(L) separates X we define L
1
= L.
Otherwise we show as in Proposition 1.4.1 that there is an (f,N)-quasi-line L
1
such that N
1
(L
1
) separates X. We continue inductively: if N
k+1
(L
k
) separates
X we define L
k+1
= L

k
otherwise we modify L
k
as in Proposition 1.4.1 to
obtain L
k+1
. We can assume that all L
k
contain the identity vertex e.
We note that by their construction the L
k
satisfy the following:
For every r>0 there is an M>0 such that for all k ≥ r every nonessential
component of X − N
r
(L
k
) is contained in N
M
(L
k
).
By passing to a subsequence we can assume that B
e
(k) ∩ L
n
does not
depend on n for n ≥ k. We define L
0
by x ∈ L

0
if x ∈ B
e
(k) ∩ L
k
. Clearly L
0
has the property required.
The following proposition shows that the essential components of
X − L have a property that one can consider as a ‘large scale’ version of
local connectedness.
Proposition 1.4.4. Let X be a Cayley graph of a finitely presented
one-ended group G.Ifan(f,N)-quasi-line L separates X then there is an
(f,N)-quasi-line L
0
which separates X and has the following property:
There is an r
0
> 0 such that for each r>r
0
there is an R>rsuch that
if d(x, L
0
)=r = d(y, L
0
), d(x, y) <f(3r) and x, y lie in the same essential
component of X − L
0
, then x, y can be joined by a path of length less than R
which does not meet L

0
.
Proof. We will show this by contradiction. Let L
0
be a separating (f, N)-
quasi-line which satisfies the following 2 properties:
1. The number of essential components of X−L
0
is the maximum possible.
2. If L
1
is a separating (f,N)-quasi-line satisfying property 1 then
sup {d(x, L
1
)}≤sup {d(x, L
0
)} where the supremum is taken over all x that
lie in a nonessential component of X − L
1
on the left side and respectively of
X − L
0
on the right side. Loosely speaking 2 just says that the nonessential
components of X − L
0
are as ‘big’ as possible.
Let r
0
be such that if d(x, L
0

) ≥ r
0
then x lies in an essential component of
X − L
0
. Suppose that L
0
does not satisfy the conclusion of the proposition for
r
0
. There are then some r>r
0
and sequences (x
n
), (y
n
) such that d(x
n
,y
n
)=
r, x
n
,y
n
lie in the same component of X − L
0
and x
n
,y

n
cannot be joined
in X − L
0
by any path of length less than n. We pick g
n
∈ G such that
g
n
x
n
= e (where e is a fixed vertex). We have then as in Proposition 1.4.1 that
QUASI-ISOMETRY INVARIANCE OF GROUP SPLITTINGS
769
a subsequence of g
n
L
0
converges on compact sets to a quasi-line L
1
such that
X − L
1
has the same number of essential components as X − L
0
. By passing if
necessary to a subsequence we have that g
n
x
n

and g
n
y
n
converge respectively
to x
0
,y
0
. Clearly x
0
,y
0
do not lie in the same essential component of X − L
1
.
It follows that at least one of them lies in a nonessential component of X − L
1
.
This however contradicts our assumption that L
0
satisfies property 2.
It is easy to see that Proposition 1.4.4 can be strengthened so that is
applies to finite neighborhoods of quasi-lines as well:
Proposition 1.4.5. Let X be a Cayley graph of a finitely presented
one-ended group G.Ifan(f,N)-quasi-line L separates X then there is an
(f,N)-quasi-line L
0
which satisfies the conclusion of Proposition 1.4.3 and has
the following property:

For any M>0 there is an r
M
> 0 such that for each r>r
M
there is an
R>rsuch that if d(x, L
0
)=r = d(y,L
0
), d(x, y) <f(3r) and x, y lie in the
same essential component of X − N
M
(L
0
), then x, y can be joined by a path of
length less than R which does not meet N
M
(L
0
).
Proof. Left to the reader.
Definition 1.5. We say that a, b ∈ X are K-separated by a quasi-line L if
d(a, L) >K,d(b, L) >Kand a ∈ X
1
,b ∈ X
2
where X
1
,X
2

are two distinct
essential connected components of X − L
It is easy to see that these notions are invariant under quasi-isometries:
Lemma 1.6. Let f : X → Y be a quasi-isometry of the geodesic metric
spaces X, Y .LetL ⊂ X be a quasi-line of X. Then there is an M>0 such
that the M-neighborhood of f(L), N
M
f(L), is a quasi-line of Y .
Proof. Left to the reader.
Lemma 1.7. Let f : X → Y be a quasi-isometry of the geodesic metric
spaces X, Y .LetL ⊂ X be a quasi-line separating X. Then there is an M>0
such that N
M
(f(L)) is a quasi-line separating Y .
Proof. Left to the reader.
Our interest in quasi-lines comes from the following:
Lemma 1.8. Let G be a finitely presented group that splits over a 2-ended
subgroup J.LetX be a Cayley graph of G. Then there is a neighborhood of J
in X that is a quasi-line separating X.
770 PANOS PAPASOGLU
Proof. Since separation is invariant by quasi-isometries we show this using
a complex naturally associated to the splitting of G (see [Sc-W]): If G =
A ∗
J
B let K
J
,K
A
,K
B

be finite complexes with π
1
(K
J
)=J, π
1
(K
A
)=A,
π
1
(K
B
)=B. We consider K
J
× [−1, 1]. Let f : K
J
→ K
A
, g : K
J
→ K
B
be
cellular maps inducing on π
1
the monomorphisms from J to A, B in G = A∗
J
B.
We glue K

J
×{−1}, K
J
×{1}, respectively to K
A
,K
B
by f, g and we obtain a
complex C with π
1
(C)=G. A similar construction applies if the splitting is an
HNN-extension. We make metric the 1-skeleton of the universal cover of C,
each
˜
C being given edge length 1. With this metric
˜
C
(1)
is quasi-isometric
to X.
If T is the Bass-Serre tree of the splitting G = A∗
J
B there is a natural map
p :
˜
C → T sending copies of
˜
K
J
× [−1, 1] to edges of T and collapsing copies

of
˜
K
A
,
˜
K
B
to vertices of T . We note that p implies distance nonincreasing. It
follows that if Z is a copy of
˜
K
J
×{0} in
˜
C,
˜
C −Z has two components, C
1
,C
2
neither of which is contained in a neighborhood of Z.
It remains to show that C
1
and C
2
are one-ended. We note that since
˜
C is
one-ended, if C

1
is not one-ended, and K is a compact set such that C
1
−K has
more than one unbounded component, then the closure in
˜
C of each unbounded
component of C
1
− K has unbounded intersection with Z. We note further
that if U is such an unbounded component of C
1
− K and a, b are two vertices
of Z lying in the closure of U and
¯
U, then there is a path in Z joining a, b
which lies in
¯
U as well. Indeed consider a path u joining a, b in U and a path
w joining them in Z. Take a Van-Kampen diagram, D, for the closed path
u ∪ w (see [L-S, Ch. 6] for a definition of Van-Kampen diagrams). Take the
maximal connected subdiagram of D containing u which maps to
¯
U. Clearly
the boundary of this subdiagram contains a path joining a, b that maps to a
path in Z. We conclude that an unbounded component of Z − K is contained
in a finite neighborhood of U.
From the discussion above it follows that in order to show that C
1
is

one-ended it suffices to prove the following:
If x is a fixed vertex in Z and if B
x
(n) is the ball of radius n centered
at x then there is a path p
n
in C
1
joining the distinct unbounded compo-
nents of Z − B
x
(n). Note that for small n, Z − B
x
(n) might have only one
unbounded component (and the condition becomes void) while for sufficiently
big n, Z − B
x
(n) has exactly two unbounded components. We show below how
to construct the paths p
n
.
We fix now a vertex of Z, x, and we consider an infinite path, q,inC
1
such that q(0) = x and such that d(q(n),Z) →∞as n →∞. We note
that a conjugate of J acts co-compactly on Z. By passing, if necessary, to an
index 2 subgroup, say J
0
, we obtain a group acting co-compactly on Z which
preserves C
1

. So there is a k>0 such that for any vertex y ∈ Z there is g ∈ J
0
such that d(gy, x) <k.
QUASI-ISOMETRY INVARIANCE OF GROUP SPLITTINGS
771
x
x + r
2
y − r
2
y
p
q
Figure 2
Given n>0 there is a vertex y ∈ Z and R>0 such that d(y, q) >n+ k
and B
y
(n + k) ⊂ B
x
(R). Since G is one-ended there is a path v joining some
vertex q(t)ofq to Z without intersecting B
x
(R). Consider now the path
q
n
= v ∪q([0,t]). Clearly d(q
n
,y) >n+ k. Let g ∈ J
0
is such that d(gy,x) <k.

It is easy to see now that we can take p
n
to be the path gq
n
.
Lemma 1.9. Let G be a finitely presented group and let X be a Cayley
graph of G.LetL be a quasi-line separating X and let Y be an essential
component of X−L. Then given r
1
> 0 there is r
2
> 0 such that any x<y∈ L
with length([x, y]
L
) > 2r
2
can be joined by a path p lying in Y ∪ L such that
a. p ∩ N
r
1
([x + r
2
,y− r
2
]
L
)=∅,
b. p ⊂ N
r
2

([x, y]
L
).
Proof. By choosing r
2
sufficiently big we can ensure that
N
r
1
([x + r
2
,y− r
2
]
L
∩ (−∞,x]
L
= ∅
and
N
r
1
([x + r
2
,y− r
2
]
L
∩ [y, ∞)
L

= ∅.
Since Y ∪L is one-ended there is a path, q, joining x, y in X−N
r
1
([x+r
2
,y−r
2
]
L
).
Let w be a path joining x, y which is contained in [x, y]
L
. We consider a Van-
Kampen diagram, D, for the closed path q ∪ w (see Figure 2).
Let f : D
(1)
→ X be the natural map from the 1-skeleton of D to X.
We remark that f
−1
((X − N
r
2
([x, y]
L
)) ∪ N
r
1
([x + r
2

,y− r
2
]
L
)) does not sep-
arate f
−1
(x) from f
−1
(y)inD. That is, there is a vertex in f
−1
(x) which
can be joined in D to a vertex in f
−1
(y) by a path which does not meet
f
−1
((X − N
r
2
([x, y]
L
)) ∪ N
r
1
([x + r
2
,y− r
2
]

L
)). Let us call this path p

. Using
the fact that L separates and that each point in L is at distance less than
N from L

we can easily modify (if necessary) p

to a path p satisfying the
conclusion of the lemma.
Convention. To translate topological arguments (like the ones in the ap-
pendix) to ‘quasi-isometric’ arguments one has to look at a space with larger
and larger scales. These scales are determined by constants that one can
explicitly compute. This is not very rewarding and so we use the following
772 PANOS PAPASOGLU
convention: We write that the statement P (r
1
,r
2
) that depends on two num-
bers r
1
,r
2
holds for r
2
 r
1
 0 if there is an R

1
> 0 such that for each
r
1
>R
1
there is an R
2
>r
1
such that for all r
2
>R
2
the statement P (r
1
,r
2
)
is true. Similarly we write for r
3
 r
2
 r
1
 0, P (r
1
,r
2
,r

3
) holds etc.
2. Separation properties of quasi-lines
The main result of this section is Proposition 2.1. It is a technical result
that will allow us to assume in the next section that quasi-lines separate X
in at most two essential components. We note that Proposition 2.1 is a ‘large
scale’ analog of a topological result (Lemma A.1 of the appendix). Its proof is a
good illustration of the techniques used in this paper, namely the ‘translation’
of topological arguments into ‘large scale geometry’ arguments.
Although the results in this section can be stated for (large scale simply
connected) metric spaces in general we will state and prove them only for
locally finite, simply connected complexes. The reason is that we are interested
in applying them to Cayley complexes of finitely presented groups.
As usual we make metric the 1-skeleton of such complexes by giving each
edge length 1, and defining the distance of two vertices to be the length of the
shortest path joining them. In what follows we will also assume that quasi-
lines are simply connected ; this is done to simplify notation. The results that
follow are valid in general for ‘large scale’ simply connected complexes as by
definition quasi-lines are ‘large scale’, simply connected.
We can always ‘fill the holes’ of a given (f, N)-quasi-line L and replace it
by a simply connected one, as long as the quasi-line is contained in the Cayley
complex of a finitely presented group G. Indeed a quasi-line is contained in
the N -neighborhood of a line L

. We join each vertex of L to a vertex of L

by
a path of length less than or equal to N. We add now to the presentation of
the group all words corresponding to simple closed curves of length less than
2N +1+f(2N + 1) in the Cayley graph of G. By this construction any closed

curve c in a quasi-line L is homotopic to a curve in L

and therefore can be
contracted to a point. Moreover there is an M>0 such that for any closed
curve in L the filling disc for c is contained in the M-neighborhood of c.In
other words the filling radius of closed curves in L is bounded by M .We
will assume in what follows that quasi-lines also have this property. We will
also assume that all separating quasi-lines considered satisfy the conclusion of
Propositions 1.4.3 and 1.4.5.
The proposition and the proof that follow give a ‘large scale analog’ of
Lemma A.1 of the appendix.
Proposition 2.1. Let X be a locally finite simply connected complex and
let L be a quasi-line separating X, such that X − L has at least three distinct
essential connected components X
1
,X
2
,X
3
. Then for any proper, increasing
QUASI-ISOMETRY INVARIANCE OF GROUP SPLITTINGS
773
f : R
+
→ R
+
and N>0 there is a K>0 such that any two vertices a, b ∈ L
that are sufficiently far apart cannot be K-separated by any (f,N) quasi-line
of X.
Proof. Suppose that L is contained in the M-neighborhood of a line L


⊂ L
(see Def. 1.3). It is clear that it suffices to show that there is K>0 such that
any two points a, b ∈ L

that are sufficiently far apart cannot be K-separated
by any (f, N)-quasi-line. Indeed this implies that any two points on L that are
sufficiently far apart cannot be K +2M-separated by any (f,N)-quasi-line.
In the argument that follows we use four constants K
1
,K,R such that
K
1
 K  R  0. It will be clear that the argument is valid if R  0,
K  R and K
1
 K. One can of course give explicit estimates (in terms of f
and N) for K
1
,K,R but we leave this to the reader.
Let a<b∈ L

so that a, b are K
1
-separated by an (f,N) quasi-line L
1
.
By Lemma 1.9 a, b can be joined in X
i
by a simple path p

i
that does not
intersect the 2K-neighborhood of the interval [a +
K
1
4
,b −
K
1
4
]

L
of L

.We
choose p
i
so that an initial and a terminal subpaths of p
i
are contained in L
while in between these subpaths p
i
does not intersect L. We call these ini-
tial and terminal subpaths, respectively, p
i0
,p
i1
and call the subpath between
them p


i
.
We consider the simple closed paths q
i
= p
i
∪ [a, b]
L

(see Figure 3) and
note that by Van-Kampen’s theorem X
i
∪ L is simply connected for all i. Let
D
i
be Van-Kampen diagrams representing a contraction of q
i
to a point inside
X
i
∪ L.
L
1
p
1
X
1
L
b

X
2
p
2
a
Figure 3
To simplify notation we denote by p
i
the subpath of ∂D
i
mapped onto p
i
.
Likewise we denote by [a, b] the subpath of ∂D
i
mapped onto [a, b]
L

.
We consider now the Van Kampen diagram D = D
1
 D
2
/ ∼ where ∼ is
given by the identification of the subpaths of D
1
,D
2
that map onto [a, b]. We
call g the natural map g : D → X sending ∂D to p

1
∪ p
2
. Let a
1
,b
1
∈ [a, b]
L

be such that the following hold:
774 PANOS PAPASOGLU
a
1
lies in the same essential component of X − L
1
as a, d(a
1
,L
1
) ≥ 2R
and a
1
is the maximal vertex in [a, b]
L

with these properties for the order
of L

.

b
1
is 2R- separated from a by L
1
and is the first vertex after a
1
on [a, b]
L

with this property.
We assume that K is chosen so that the following holds: There is at most one
interval, [x, y]
L

1
of L

1
with the following properties:
a. [x, y]
L
1
intersects [a
1
,b
1
]
L
.
b. B

N
(x) and B
N
(y) intersect g(∂D).
c. If z ∈ (x, y)
L

1
then B
N
(z) does not intersect g(∂D).
We will show now that there is an interval [x, y]
L

1
with the above properties
such that [x, y]
L
1
contains a path joining B
N
(x)toB
N
(y) which is contained in
X
1
∪X
2
. We assume that no such interval exists and we argue by contradiction.
Consider all maximal subdiagrams of D,sayU, with the property that ∂U

is in g
−1
(L
1
). Since L
1
is simply connected we can modify all such U so
that U ⊂ g
−1
(L
1
) (we cut away all such diagrams U and glue back diagrams
contracting ∂U to a point inside L
1
).
We consider now the connected components of D − g
−1
(L). Let V
1
be the
component containing p

1
and V
2
be the component containing p

2
. Let w be a
simple path in ∂V

1
separating p

1
from p

2
. Then g(w) is contained in L.IfI
is a minimal interval of L containing g(w) then one sees easily that I contains
[a +
K
1
4
,b−
K
1
4
]
L
. We conclude that g(w) intersects both B
R
(a
1
),B
R
(b
1
).
Let a


1
,b

1
be such that g(a

1
) ∈ B
R
(a
1
), g(b

1
) ∈ B
R
(b
1
). Then a

1
,b

1
are
separated in D by a path, say q, such that g(q) ⊂ L
1
and ∂q ⊂ ∂D.Now,∂q
separates ∂D in two paths, say c
1

,c
2
. Clearly neither c
1
nor c
2
is contained
in L
1
. We consider the shortest path in L
1
, with the same endpoints as q,
which is contained in X
1
∪ X
2
. For convenience we still call this path q.
We consider Van-Kampen diagrams for c
1
∪ q and c
2
∪ q. Since q does not
intersect [a
1
,b
1
]
L
one of either c
1

∪ q or c
2
∪ q has the property that any Van-
Kampen diagram corresponding to it contains two points, say a

1
,b

1
, such that
g(a

1
) ∈ B
R
(a
1
), g(b

1
) ∈ B
R
(b
1
). Let us say this is the case for c
1
∪ q. We pick
a Van-Kampen diagram for c
1
∪ q, and repeat the procedure. This is bound to

stop after finitely many steps, producing a diagram in which the preimage of
B
R
(a
1
) is not separated from the preimage of B
R
(b
1
) by the preimage of L
1
,
a contradiction.
We showed therefore that there exists an interval [x, y]
L

1
of L

1
with the
properties a,b,c described above such that [x, y]
L
1
contains a path joining
B
N
(x)toB
N
(y) which is contained in X

1
∪ X
2
.
QUASI-ISOMETRY INVARIANCE OF GROUP SPLITTINGS
775
By considering D
1
,D
3
we see that [x, y]
L
1
contains another path joining
B
N
(x)toB
N
(y) which is contained in X
1
∪ X
3
. This implies that B
N
(x),
B
N
(y) are both contained in X
1
. On the other hand by considering D

2
,D
3
we
conclude that B
N
(x), B
N
(y) are both contained in X
2
, a contradiction. This
proves Proposition 2.1.
3. A geometric characterization of virtually planar groups
In this section we give a quasi-isometric characterization of virtual surface
groups. It is modeled after Theorem A of the appendix and its proof follows
closely the proof of this theorem. Roughly what we show is that if G is a
one-ended finitely presented group such that any two points in its Cayley
graph which are sufficiently far away are separated by a quasi-line, then G
is virtually a surface group. In the proof of Theorem A one shows that every
simple closed curve separates and uses a classical theorem of Bing [Bi] to
conclude the proof. For Theorem 3.1 below we distinguish two cases: in the
nonhyperbolic case we show also that appropriately chosen (with big filling
radius) ‘thickened’ simple closed curves separate while in the hyperbolic case
we show that ‘thickened’ geodesics separate. In the first case we conclude using
Varopoulos’ isoperimetric inequality and in the second using the Tukia-Gabai
theorem on convergence groups. To show that thickened simple closed curves
separate we argue as for Theorem A: We use separating quasi-lines to define
what it means for a point to be ‘inside’ a simple closed curve. (see the definition
after Lemma 3.2.1). The main technical result is Lemma 3.2 which parallels
Lemma A.3.3 of the appendix. This is used to show later in Lemma 3.4 that

the definition of ‘inside’ does not depend essentially on the quasi-line picked.
We state now the main result of this section:
Theorem 3.1. Let G be a one-ended group and let X = X
G
be a Cayley
complex of G. Suppose that there is a proper increasing f : R
+
→ R
+
and
N>0 such that for any K>0 there is an M>0 such that any two points
x, y of X with d(x, y) >M are K-separated by some (f,N) quasi-line. Then
G is commensurable to a fundamental group of a surface.
Proof. In the proof that follows we suppose that K  N. It will be clear
that the argument is valid for K sufficiently bigger than N, and it is easy to
obtain an explicit estimate for K. We will need some technical lemmas:
Let L be a quasi-line separating X and let L

be its corresponding line. By
Proposition 2.1 X−L contains exactly two essential connected components. We
denote them respectively L
+
,L
−
. We denote
¯
L
+
the union L
+

∪ L. Similarly
¯
L
−
= L
−
∪ L.
The next lemma is the ‘large scale’ analog of Lemma A.3.3 of the appendix.
776 PANOS PAPASOGLU
Lemma 3.2. Let L
1
, L
2
be separating (f,N) quasi-lines and let L

1
,L

2
be
the corresponding lines. For r
2
 r
1
 0 the following hold:
Let a, b ∈ L

1
be such that a, b ∈ L
−

2
, d(a, L
2
) ≥ r
1
,d(b, L
2
) ≥ r
1
and for
all t ∈ (a, b)
L

1
, d(t,
¯
L
+
2
) <r
1
.LetI be a minimal interval of L
2
containing
[a, b]
L
1
∩ L
2
.

Let x ∈ I ∩ L

2
, y ∈ (L
2
− I) ∩ L

2
be such that d(x, L
1
) >r
2
, d(y, L
1
) >r
2
.
Then any path p joining x, y and lying in
¯
L
+
2
intersects [a, b]
L
1
.
Proof. The proof is similar to that of Lemma A.3.3 of the appendix. We
need a lemma similar to Lemma A.3.3.1 of the appendix:
Lemma 3.2.0. With the notation of Lemma 3.2 the following holds:
Let S =([a, b]

L
1
∩L
2
)−([a, b]
L
1
∩[x, y]
L
2
). There is a path p

in
¯
L
−
2
joining
x, y and a Van-Kampen diagram g :D →X for p

∪ [x, y]
L

2
such that:
a) d(g(D),S) >r
1
.
b) d(p


, [a, b]
L
1
∩ [x, y]
L
2
) >r
1
.
Proof. Let p
1
be a path in
¯
L
−
2
joining x, y such that d(p
1
, [a, b]
L
1
)
> 3r
1
. We note that such a path exists by our assumption that r
2
 r
1
and Lemma 1.9. Let g
1

: D
1
→ X be a Van-Kampen diagram for p
1
∪ [x, y]
L

2
.
Let s ∈ g
−1
1
(S) and let O
s
be the component of s in D
1
−g
−1
1
(N
r
1
([x, y]
L

2
)∩L
2
).
If ∂O

s
⊂ g
−1
1
(N
r
1
([x, y]
L

2
) ∩ L
2
) we modify D
1
as follows: Since ∂O
s
⊂ L
2
we
fill ∂O
s
in L
2
, so after the change g
1
(O
s
) ⊂ N
2r

1
([x, y]
L

2
) ∩ L
2
.
By performing this ‘cut and paste’ operation for all s ∈ g
−1
1
(S)asabove
we get a Van-Kampen diagram, that we still call D
1
, such that all s ∈ g
−1
1
(S)
belong to a single component of D
1
− g
−1
1
(N
r
1
([x, y]
L

2

) ∩ L
2
).
We consider now the subdiagram D
r
1
of D
1
consisting of all closed 2-cells
σ of D
1
such that d(g
1
(σ), [x, y]
L

2
) ≤ 2r
1
. Let D

1
be the connected component
of D
r
1
containing [x, y]
L

2

. Clearly there is a path p
2
∈ ∂D

1
joining x, y such
that d(g(p
2
), [a, b]
L
1
∩ [x, y]
L
2
) >r
1
. We then take p

= g
1
(p
2
). Clearly if we
take D to be the subdiagram of D
1
bounded by [x, y]
L

2
∪ p

2
and g = g
1
|
D
both
conditions a) and b) are satisfied.
We return now to the proof of Lemma 3.2 arguing by contradiction. Let
p be a path joining x, y in
¯
L
+
2
such that p does not intersect [a, b]
L
1
. Let
h
1
: E
1
→ X be a Van-Kampen diagram for p∪[x, y]
L

2
such that h
1
(E
1
) ⊂

¯
L
+
2
.
Let h : E → X be the Van-Kampen diagram obtained by identifying D,E
1
along [x, y]
L

2
.
Let r
0
be such that r
1
 r
0
 0. By Remark 3.3, there are z
1
,z
2
∈ [x, y]
L

2
such that z
1
,z
2

are separated by L
1
and d(z
1
, [a, b]
L
1
)=d(z
2
, [a, b]
L
1
)=2r
0
.
Let t
1
∈ h
−1
(N
r
0
(z
1
)),t
2
∈ h
−1
(N
r

0
(z
2
)). Now, t
1
,t
2
are separated by h
−1
(L
1
).
Therefore there is a minimal simple path c in h
−1
(L
1
) separating t
1
,t
2
.Ifc is
QUASI-ISOMETRY INVARIANCE OF GROUP SPLITTINGS
777
a closed path then either t
1
or t
2
is contained in the region it bounds. We can
eliminate then one of t
1

,t
2
by cutting the region bounded by c and gluing back
a Van-Kampen diagram for c whose image is contained in L
1
.Ifc is not closed
its endpoints are contained in ∂E.Ifc
1
,c
2
are the endpoints of c then both
c
1
,c
2
∈ p

and (by Lemma 3.2.0) [h(c
1
),h(c
2
)]
L
1
does not intersect [z
1
,z
2
]
L

2
.
We cut E open along c−{c
1
,c
2
} and obtain thus a diagram with a region,
say F , in its interior that is bounded by two copies of c. We join c
1
,c
2
in F
by a path ¯c. We extend h to ¯c by mapping it to a path lying in [h(c
1
),h(c
2
)]
L
1
that joins h(c
1
),h(c
2
). F is subdivided by ¯c in two regions. Each region is
bounded by ¯c ∪ c. We fill this regions by Van-Kampen diagrams that contract
h(¯c ∪ c) to a point inside [h(c
1
),h(c
2
)]

L
1
. We obtain thus a diagram, that we
still call E for convenience, in which t
1
,t
2
are separated by ¯c. We remark that
h(¯c) does not intersect [z
1
,z
2
]
L
2
.
It is easy to see that repeating this ‘cut and paste’ operation finitely many
times we obtain a Van-Kampen diagram h : E → X, for p ∪ p

such that the
following holds: If t
1
∈ h
−1
(N
r
0
(z
1
),t

2
∈ h
−1
(N
r
0
(z
2
) then t
1
,t
2
are separated
in E by a simple path c ∈ h
−1
(L
1
) such that h(c) does not intersect [z
1
,z
2
]
L
2
.
This is clearly impossible.
Lemma 3.2 holds also for infinite intervals. More precisely we have the
following:
Lemma 3.2.1. Let L
1

, L
2
be separating (f,N) quasi-lines and let L

1
,L

2
be the corresponding lines. For r
2
 r
1
 0 the following holds:
Let a, c ∈ L

1
be such that a ∈ L
−
2
, c ∈ L
+
2
and d(a, L
2
) ≥ r
1
,d(c, L
2
) ≥ r
1

and for all t ∈ (a, ∞]
L

1
d(t,
¯
L
+
2
) <r
1
.Letb ∈ [a, c]
L
1
∩ L
2
and let I
1
,I
2
be the
infinite connected components of L
2
− B
r
1
(b).
Let x ∈ I
1
, y ∈ I

2
be such that d(x, L
1
) >r
2
, d(y,L
1
) >r
2
. Then any
path p joining x, y and lying in
¯
L
+
2
intersects [a, ∞)
L
1
.
Proof. Left to the reader.
The following definition is similar to Definition A.4 of the appendix:
Definition. Let C be a closed curve in X and let L be an (f,N) quasi-
line. Let x ∈ L be such that d(x, C) >R. We say that a subpath of C lying
in L
+
(or in L
−
)isR-above x if the following are satisfied:
1) ∂I ⊂ L.
2) x lies in the interval of L determined by ∂I.

3) I is a maximal subpath satisfying 1), 2).
We say that x is an (R, L)-interior point of C if there is an odd number of
subpaths of C in L
+
that are R-above x.
778 PANOS PAPASOGLU
Note that for sufficiently big R this does not depend on our choice of line
L

for L. When such an R is given we say simply that x is an L-interior point
of C an (rather than an (R, L)-interior point).
To state the next lemma we need a definition:
Definition. Let C be a simple closed curve. We say that C is locally
(c
1
,c
2
)-quasigeodesic if every subpath of C of length less than length(C)/2is
a (finite) (c
1
,c
2
)-quasigeodesic.
The next lemma is an analogue of Lemma A.4.2 of the appendix:
Lemma 3.3. Let c
1
,c
2
,R > 0 be given. For any sufficiently big R
1

>R
the following holds: Let C be a simple closed curve that is a locally (c
1
,c
2
)
quasi-geodesic and let L be a separating quasi-line of X, R
1
-separating a, b ∈ C.
Then there is an x ∈ L such that x is an (R, L)-interior point of C.
The proof, left to the reader, is the same as the proof of Lemma A.4.2 of
the appendix.
We need a definition:
Definition. We say that a quasi-line L
1
, r-crosses a quasi-line L
2
at
[x, y]
L
1
if x, y are r-separated by L
2
. We also say that [x, y]
L
1
r-crosses L
2
.
Remark 3.3. As in Lemma A.2 of the appendix we remark that there is

a proper function g : R
+
→ R
+
such that if L
1
r-crossses L
2
at [x, y]
L
1
then
L
2
g(r)-crosses L
1
at an interval [a, b]
L
2
contained in the r-neighborhood of
[x, y]
L
1
.
The following lemma is an analogue to Lemma A.4.3 of the appendix.
Lemma 3.4. Let C be a simple closed curve in X and let L
1
, L
2
be sep-

arating quasi-lines. For any sufficiently big r>0 and R  r the following
holds: Let x<ybe points on L

1
such that x is r-separated from y by L
2
. Sup-
pose that d([x, y]
L
1
,C) >R.Ift ∈ [x, y]
L
1
∩ L
2
then t is an (R, L
1
)-interior
point of C if and only if it is an (R, L
2
)-interior point of C.
Proof. The proof is similar to the proof of Lemma A.4.3 of the appendix.
Some modifications however have to be made since L
1
, L
2
are quasi-lines and
not lines, and so it is not possible to have a ‘planar’ picture of L
1
, L

2
such that
points of L
1
(or L
2
) separated by L
2
are mapped on the plane to points that
are separated by the image of L
2
. What one can show roughly is that points
on L
1
(or L
2
) ‘far away’ from L
2
(or L
1
) are correctly mapped to the plane.
We explain this here in some detail.
QUASI-ISOMETRY INVARIANCE OF GROUP SPLITTINGS
779
We define first a map from L

1
L

2

to the plane. Mapping L

2
to the x-axis
in a length-preserving way, for convenience, we identify L

2
to its image. We
assume r  r
2
 r
1
where r
1
,r
2
are such that lemma 3.2 holds. To each
minimal interval [x
1
,x
2
]
L

1
of L
1
such that [x
1
,x

2
]
L
1
r
1
-crosses L
2
we associate
a point lying in [x
1
,x
2
]
L
1
∩ L

2
.
Let J =[a, b]
L

1
be an interval of L

1
satisfying the conditions of Lemma 3.2.
Using J we will define a map from a smaller interval contained in J to the plane.
Let [a, a

1
]
L

1
and [b
1
,b]
L

1
be minimal intervals such that [a, a
1
]
L
1
and [b
1
,b]
L
1
r
1
-cross L
2
. Let a
2
,b
2
be the points on L


2
corresponding to these intervals.
Let a

2
,b

2
∈ L

1
be points on L

1
such that d(a
2
,a

2
) ≤ N, d(b
2
,b

2
) ≤ N (where
a

2
,b


2
are obtained by the usual map from the quasi-line L
1
to the line L

1
).
We map then [a

2
,b

2
]
L

1
to a polygonal path joining a
2
,b
2
. This polygonal
path intersects the x-axis only at its endpoints and lies in the half-plane y>0
if a, b ∈ L
−
2
and in the half-plane y<0ifa, b ∈ L
+
2

.
In a similar way, we map infinite intervals of L

1
using intervals of the form
[a, ∞)
L

1
or of the form (−∞,a]
L

1
satisfying the conditions of Lemma 3.2.1.
We note now that we can write L

1
as a union of intervals satisfying the
conditions of either Lemma 3.2 or 3.2.1. Two such intervals J
1
,J
2
can intersect
‘near’ their endpoints. There is a natural order on this set of intervals of L

1
as
each interval intersects exactly one other interval near each of its endpoints.
Call two intervals that intersect adjacent. If two intervals satisfying 3.2 or
3.2.1 are adjacent then using these intervals we define a map from two intervals

contained in them. These two new intervals intersect at exactly one point and
are mapped to polygonal lines which also intersect at exactly one point. We
can, therefore, using these maps define a map from L

1
to the plane; each point
of L

1
belongs to an interval [a

2
,b

2
]
L

1
as above and is mapped to the plane by
the corresponding map. We call g : L

1
→ R
2
the map obtained in this way
and note that g might not be one-to-one. It is possible that two intervals,
say [a, b]
L


1
,[c, d]
L

1
, and are mapped to the plane so that g(a),g(b),g(c),g(d)
lie on the x-axis, (a, b)
L

1
,(c, d)
L

1
are mapped both, either in the upper or in
the lower half-plane and g(c) ∈ [g(a),g(b)] while g(d) /∈ [g(a),g(b)]. In this
case the image of [a, b]
L

1
intersects the image of [c, d]
L

1
. By changing the map
g if necessary we can assume that g([a, b]
L

1
) intersects g([c, d]

L

1
) at exactly
one point. We can further assume that for any pair of intervals as above for
which g(c),g(d) are either both inside or both outside [g(a),g(b)] the images
of [a, b]
L

1
,[c, d]
L

1
do not intersect.
We explain now how to modify g so that g(L

1
) is a line. We fix an interval
[a, b]
L

1
and consider all intervals such that their images intersect g([a, b]
L

1
). We
say g([c, d]
L


1
) intersects g([a, b]
L

1
), g(a),g(b),g(c),g(d) lie on the x-axis, and
(a, b)
L

1
,(c, d)
L

1
are both mapped in the upper half-plane. Let I be the bounded
interval of L

1
− ((a, b)
L

1
∪ (c, d)
L

1
). I then joins an endpoint of [c, d]
L


1
to an
endpoint of [a, b]
L

1
. To fix ideas let us say that the endpoints of I are a, c.We
780 PANOS PAPASOGLU
change g so that g(I)=g(a) and so that g([c, d]
L

1
) becomes a polygonal path
in the upper half-plane joining g(d)tog(a). We pick this polygonal path so
that it does not intersect g((a, b]
L

1
) and any other paths of g(L

) in the upper
half-plane that have both their endpoints either inside or outside [g(a),g(d)].
If r
3
 r
2
we have that length(I) <r
3
by Lemmas 3.2, 3.2.1. This mod-
ification is made for every interval [c, d]

L

1
whose image intersects g((a, b)
L

1
).
In this way eventually g((a, b)
L

1
) does not intersect any other polygonal path.
We continue by picking another interval and changing the map in the same
away to eliminate intersections with the image of this interval. As there are
countably many intervals, it is clear that we can eliminate all self -intersections
of g(L

1
). Note that after these modifications some intervals I of L

1
are mapped
to a point but all such intervals have length smaller than r
3
.
For r
3
big enough one can verify easily that if a, b ∈ L


1
are such that
d(a, L
2
) >r
3
, d(b, L
2
) >r
3
, then a, b are r
3
-separated by L
2
if and only if
g(a),g(b) are separated by g(L

2
).
The next lemma shows that this holds also for a, b ∈ L

2
:
Lemma 3.4.1. If a, b ∈ L

2
are such that d(a, L
1
) >r
3

, d(b, L
1
) >r
3
then
a, b are r
3
-separated by L
1
if and only if g(a),g(b) are separated by g(L

1
).
Proof. We show first that if a, b ∈ L

2
are not separated by L
1
then
g(a),g(b) are not separated by g(L

1
). We note that there is a path p joining
a, b in X that does not intersect the r
2
-neighborhood of L
1
. Indeed, if this
were not so, X − N
r

2
(L
1
) would have more that two essential components (see
Def. 1.4). This contradicts the hypothesis of Theorem 3.1 (see Prop. 2.1). We
decompose p as a union p = p
1
∪···∪p
n
where the p
i
’s are successive subpaths
lying in
¯
L
+
2
or in
¯
L
−
2
. Let a
i
,b
i
be the endpoints of p
i
. We fix some p
i

and
suppose that, say, p
i
⊂
¯
L
+
2
. Note that if there is no path joining the points
g(a
i
),g(b
i
)inR
2
− g(L

1
) then a
i
,b
i
are separated in
¯
L
+
2
by an interval of L
1
as in Lemma 3.2 (or in Lemma 3.2.1). This is impossible. We conclude that

g(a),g(b) are not separated by g(L

1
).
Similarly, if for a, b as in the lemma, g(a),g(b) are not separated by g(L

1
)
then it is easy to see that g(a),g(b) can be joined by a path that does not meet
g(N
r
2
(L
1
)). This in turn implies that a, b can be joined by a path that does
not meet L
1
.
Note that g(L

1
) and g(L

2
) separate the plane in two pieces. We define
g(L
+
1
) to be the component of R
2

− g(L
1
) which contains g(a) for some a ∈ L

2
such that a ∈ L
+
1
and d(a, L
1
) >r
3
. We define similarly g(L
−
1
),g(L
+
2
),g(L
−
2
),
and now extend g to C, so that g will be defined on L
1
 L
2
 C.
Assuming that r  r
3
, we show how to map C to the plane so that

t ∈ [x, y]
L

1
∩ L
2
is an (R, L
1
)-interior point ((R, L
2
)-interior point) of C if and
only if g(t)isag(L

1
)(g(L

2
) ) interior point of the image of C.
QUASI-ISOMETRY INVARIANCE OF GROUP SPLITTINGS
781
We extend g to L
1
∪L
2
in the obvious way by defining g(a)=g(a

), where
a → a

is the usual map from a line to the corresponding quasi-line.

We decompose C as a union of successive paths C = C
1
∪···∪C
n
where
the endpoints of C
i
are on L
2
and C
i
is contained in
¯
L
+
2
or in
¯
L
−
2
. We explain
now how to map each C
i
to the plane. Let us say that the endpoints of C
i
are
a
i
,b

i
and, to fix ideas, suppose that C
i
is contained in
¯
L
+
2
. We will map C
i
to
the plane so that the following conditions hold:
1) If a ∈ C
i
∩ L
1
then g(a) ∈ g(C
i
).
2) If for some a ∈ L
1
, g(a) ∈ g(C
i
), then d(a, C
i
) <r
3
.
We orient C
i

from a
i
to b
i
. Let c
1
, ,c
r
be the vertices of intersection of C
i
with L
1
in the order they appear. If c
1
∈ L
+
2
we map the subpath [a
i
,c
1
]ofC
i
to be a polygonal line in g(L
+
2
) joining g(a
i
)tog(c
1

) and having the minimum
possible number of intersection points with g(L
1
). Note that any intersection
point of g([a
i
,c
1
]) with g(L
1
) lies in an interval of g(L
1
) that lies in g(L
+
2
) and
separates a
i
,c
1
. By the definition of g and Lemma 3.2 one sees easily that such
an interval contains a point g(t) such that either d(t, a
i
) <r
3
or d(t, c
1
) <r
3
.

We choose g([a
i
,c
1
]) so that it intersects this interval exactly at a point g(t)
with the above property.
If c
1
∈ L
−
1
then we pick a point s on L
2
such that d(s, c
1
) <r
3
and we
define g([a
i
,c
1
]) to be the union of three polygonal paths: the first joins g(a
i
)
to g(s) and lies in g(L
+
2
); the second joins g(s)tog(c
1

) and lies in g(L
−
2
);
and the third is the inverse of the second. This path might intersect g(L
1
)at
points other than c
1
. We can however arrange, as before, that if g(t) is such
an intersection point then either d(t, a
i
) <r
3
or d(t, c
1
) <r
3
.
We continue in the same way defining g([c
1
,c
2
]) to be a polygonal path
joining g(c
1
),g(c
2
) and lying in g(L
+

2
)ifg(c
1
),g(c
2
) lie in g(L
+
2
). If one of
them or both lie in g(L
−
2
) then we define this path as before using an auxiliary
point on g(L
2
) close to the point in g(L
−
2
). We define g on all subpaths C
i
in
the same way.
By the remarks made above we can arrange so that g satisfies the following:
If t ∈ C is such that g(t) ∈ g(C)∩g(L
1
)org(t) ∈ g(C)∩g(L
2
) then, respectively,
d(t, L
1

) <r
3
or d(t, L
2
) <r
3
.
Note that x is an L
2
interior point of C if and only if g(x)isag(L
2
)
interior point of g(C). Indeed this follows easily from the definition of g, the
extra intersection points with g(L
2
) that we might create defining g(C) do not
change the parity of intervals above g(x).
We will show now that x is an L
1
interior point of C if and only if g(x)
is a g(L
1
) interior point of g(C). Let p =[b
1
,b
2
] be a subpath of C lying in
¯
L
+

1
with b
1
,b
2
∈ L
1
. Then p can be written as a union of successive subpaths
p = p
1
∪···∪p
n
such that for each i one of the following two holds:
782 PANOS PAPASOGLU
1) g(p
i
) ⊂ R
2
− g(L
1
) and the endpoints of g(p
i
) lie on g(L
1
) and are sepa-
rated on g(L
1
)byg(x), or
2) g(p
i

) ⊂ R
2
− g(L
1
) and the endpoints of g(p
i
) lie on g(L
1
) and are not
separated on g(L
1
)byg(x).
We will show that in case 1) g(p
i
) lies in g(
¯
L
+
1
). Let q be a subpath of p
i
satisfying the following:
Each endpoint of g(q) lies either on g(L
1
)orong(L
2
), g(q) is contained
either in g(
¯
L

+
2
)oring(
¯
L
−
2
) and there is a point t on q at distance bigger than
r
3
from L
1
. To fix ideas we assume that g(q) lies in g(
¯
L
+
2
). We then join t to a
point s on L
2
by a path that meets L
2
only at s and does not intersect the r
2
neighborhood of L
1
. Clearly s ∈ L
+
1
. From Lemmas 3.2, 3.2.1 it follows that

g(s) and the endpoints of g(q) lie in the same component of g(
¯
L
+
2
)−g(L
1
). This
implies that g(q) is at the same component of R
2
− g(L
1
)asg(s). Therefore
g(p
i
) ⊂ g(
¯
L
+
1
).
From these observations it follows that the number of subpaths of g(p)
lying above g(x)ing(L
+
1
) is odd. This in turn implies that g(x)isag(L
1
)
interior point for g(C) if and only if it is an L
1

-interior point for C. Since g(x)
is a g(L
1
) interior point for g(C) if and only if it is an g(L
2
) interior point for
g(C) the lemma follows.
We show now how to approximate any path in X by “polygonal paths”
i.e. paths made by intervals of quasi-lines. This is similar to Lemma A.4.4 of
the appendix.
Remark 3.5. Note that for any r>0 there is an R>rsuch that for
any separating (f,N)-quasi-line L the following holds: If O ∈ X − N
r
(L) and
d(O, L) >Rthen O lies in an essential component of X − N
r
(L).
This is because (see §2) all separating quasi-lines considered satisfy the
conclusion of Proposition 1.4.3.
Lemma 3.6. For any r>0 there is an R>rsuch that for any separating
(f,N)-quasi-line L the following holds: Let x, y ∈ X be such that d(x, L)=
d(y, L)=r +1 and let x

,y

∈ L be such that d(x, x

)=d(y, y

)=r +1.Ifx, y

lie in the same essential component of X − N
r
(L) then x, y can be joined by a
path lying in (X − N
r
(L)) ∩ N
R
([x

,y

]
L
.
Proof. The lemma follows easily from Proposition 1.4.5.
To construct the “polygonal paths” needed we use some constants M,R,r.
In what follows we suppose that M  R  r  0. We describe below some
of the properties of these constants.
Suppose that r>Nis big enough so that Lemma 3.4 holds. We take R big
enough so that Remark 3.5 and Lemma 3.6 hold for R/4 (where r is as given